Abstract
Formal verification technology has been widely applied in the fields of mathematics and computer science. The formalization of fundamental mathematical theories is particularly essential. Axiomatic set theory is a foundational system of mathematics and has important applications in computer science. Most of the basic concepts and theories in computer science are described and demonstrated in terms of set theory. In this paper, we present a formal system of axiomatic set theory based on the Coq proof assistant. The axiomatic system used in the formal system refers to Morse-Kelley set theory which is a relatively complete and concise axiomatic set theory. In this formal system, we complete the formalization of the basic definitions of sets, functions, ordinal numbers, and cardinal numbers and prove the most commonly used theorems in Coq. Moreover, the non-negative integers are defined, and Peano’s postulates are proved as theorems. According to the axiom of choice, we also present formal proofs of the Hausdorff maximal principle and Schroeder-Bernstein theorem. The whole formalization of the system includes eight axioms, one axiom schema, 62 definitions, and 148 corollaries or theorems. The “axiomatic set theory” formal system is free from the more apparent paradoxes, and a complete axiomatic system is constructed through it. It is designed to give a foundation for mathematics quickly and naturally. On the basis of the system, we can prove many famous mathematical theorems and quickly formalize the theories of topology, modern algebra, data structure, database, artificial intelligence, and so on. It will become an essential theoretical basis for mathematics, computer science, philosophy, and other disciplines.
Highlights
With the rapid development of computer science, especially the emergence of interactive theorem proving tools Coq [1], [2], Isabelle/HOL [3] and so on, the formal verification technology has made excellent progress [4], [5]
Wiedijk pointed out that relevant research groups around the world have completed or plan to complete the formal proof of 100 famous mathematical theorems including Gödel incompleteness theorem, Prime number theorem, The associate editor coordinating the review of this manuscript and approving it for publication was Engang Tian
Zhong Shao, and others launched the Deep Specification project which focuses on the specification and verification of full functional correctness of software and hardware in 2016 [11]. These results show that formal verification technology plays an essential role in the fields of mathematics and computer science
Summary
With the rapid development of computer science, especially the emergence of interactive theorem proving tools Coq [1], [2], Isabelle/HOL [3] and so on, the formal verification technology has made excellent progress [4], [5]. Set_Theory/ The formal system can quickly build a foundation for mathematics and present a concise and relatively complete axiomatic set theory in Coq. Compared with naive set theory in Coq standard library, the system has many advantages. We formalize some basic definitions of sets, functions, ordinal numbers, integers, and cardinal numbers and prove the most commonly used theorems in this formal system. We present a formal proof of the mathematical induction by the definition of integers and propose some properties related to finiteness. Appendix lists the formal description of essential theorems in the system
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